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Dressing with Control: Using Integrability to Generate Desired Solutions to Einstein’s Equations

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The sine-Gordon Model and its Applications

Part of the book series: Nonlinear Systems and Complexity ((NSCH,volume 10))

Abstract

Motivated by integrability of the sine-Gordon equation, we investigate a technique for constructing desired solutions to Einstein’s equations by combining a dressing technique with a control-theory approach. After reviewing classical integrability, we recall two well-known Killing field reductions of Einstein’s equations, unify them using a harmonic map formulation, and state two results on the integrability of the equations and solvability of the dressing system. The resulting algorithm is then combined with an asymptotic analysis to produce constraints on the degrees of freedom arising in the solution-generation mechanism. The approach is carried out explicitly for the Einstein vacuum equations. Applications of the technique to other geometric field theories are also discussed.

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Notes

  1. 1.

    The term integrable has been adopted in various contexts, and is ambiguous in characterizing the features of such an equation (e.g., existence of closed-form solutions by quadrature, possessing infinitely many conservation laws, exhibiting solitonic dynamics, etc.). We shall restrict our attention to those equations which are integrable in the Lax sense.

  2. 2.

    It is a long-standing open problem to characterize the PDEs which admit a Lax formulation.

  3. 3.

    It also hinges on space being non-compact and one dimensional.

  4. 4.

    Although one is no longer considering an eigenvalue problem in the classical sense, λ is often still called a spectral parameter.

  5. 5.

    Using the transformation \(x =\zeta +\eta,t =\zeta -\eta\), one may easily recover the second recognizable form of the sine-Gordon equation, \(u_{\mathit{xx}} - u_{\mathit{tt}} =\sin u\).

  6. 6.

    Asymptotically flat solutions represent vacuum outside an isolated body.

  7. 7.

    The sine-Gordon equation can also be put into this framework, see [22].

  8. 8.

    The metric is Riemannian if G is a compact group.

  9. 9.

    Note that by virtue of the Cartan embedding, a symmetric space can be embedded into a Lie group, for which one may choose a matrix representation, thus making addition and scalar multiplication of elements possible.

  10. 10.

    In particular the pseudo-unitary groups SU(p, q) satisfy this assumption. Moreover, groups possessing Dynkin diagrams with no symmetries may also qualify. Modification of this procedure to address other families of involutions will be taken up in a future work.

  11. 11.

    Note however, that all of the known uniqueness results for the Kerr metric are in the black hole regime, and cover only the outside of the event horizon (cf. [40] and references therein.) The same holds for “double-Kerr” non-existence results (e.g. [41, 42]).

  12. 12.

    The first two columns of Table 1 are excerpted directly from [38], except the fifth row (see [43]).

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Acknowledgements

SB gratefully acknowledges support from M. Kiessling and the NSF through grant DMS-0807705, during which the first part of this project was conceived; this material is also based upon work supported by the NSF under Grant #0932078000, while SB was in residence at the Mathematical Science Research Institute in Berkeley, California, during the 2013 Autumn semester. STZ thanks the Institute for Advanced Study for their hospitality and the stimulating environment provided during Spring 2011 while the authors were working on this project.

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  1. Department of Mathematics, Rutgers, The State University of New Jersey, Piscataway, NJ, 08854, USA

    Shabnam Beheshti & Shadi Tahvildar-Zadeh

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  1. Shabnam Beheshti
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Correspondence to Shabnam Beheshti .

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  1. Departamento de Físíca Aplicada I, University of Sevilla, Sevilla, Spain

    Jesús Cuevas-Maraver

  2. Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA

    Panayotis G. Kevrekidis

  3. Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts, USA

    Floyd Williams

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Beheshti, S., Tahvildar-Zadeh, S. (2014). Dressing with Control: Using Integrability to Generate Desired Solutions to Einstein’s Equations. In: Cuevas-Maraver, J., Kevrekidis, P., Williams, F. (eds) The sine-Gordon Model and its Applications. Nonlinear Systems and Complexity, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-06722-3_9

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  • DOI: https://doi.org/10.1007/978-3-319-06722-3_9

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